I've heard and used both and would never dream of saying one didn't exist when both obviously do.
However, to me 'edgewise' is more standard or formal English, and more common in the idiom with 'get a word in.' I think of 'edgeways' as being much more colloquial or regional, and used mainly in the literal sense, for positioning an object. It's surprising to me that it's apparently more common in BE even in contexts such as journalism.
But as most of us have had ample occasion to recognize, you learn something new every day.
The OED lists both under edgewise, -ways, infrequently -way; the earliest entry, IIRC, was for -waie (= -way), followed by one or two for -ways, but by the 18th century -wise seemed to have become more common.
Fowler/Burchfield has a nice article under -wise, -ways that's too long to quote here, but with many examples of words with both (which would also be good to double-check in LEO). One interesting quote:
Dr. Johnson (in his Dictionary s.v. Way 25) ... said that 'Way and ways are now often used corruptly for -wise', and this statement, as the OED points out, 'has probably led many to prefer -wise to -ways or -way on the ground of supposed correctness.
So perhaps the preferred usage has swung back in BE, but not in AE. Garner has
edgewise; edgeways. Both words are mostly restricted to variations on the phrase couldn't get a word in edgewise. In AmE, it's always edgewise; in BrE, it's often edgeways.